Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 18 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii
- iii a
  - Solution iii

i

Solution

A standard complete orthogonal family is the real Fourier basis

1, cos x, sin x, cos 2 x, sin 2 x, \dots .

Completeness gives expansion in the L2 space for a square-integrable function. Mere integrability alone does not guarantee pointwise convergence of a Fourier series; the explicit function in part (b) is continuous and piecewise smooth, so this issue does not obstruct its expansion.

ii

Solution

With inner product

⟨ u, v ⟩ = \int_{- π}^{π} u (x) v (x) d x

, orthogonality means

\int_{- π}^{π} e_{n} (x) e_{m} (x) d x = 0 (m \neq = n) .

The constant function has squared norm

2 π

, and each sine or cosine has squared norm

π

. Thus the displayed family is orthogonal but not normalized; these different lengths determine the coefficient factors.

iii

Solution

The standard Fourier series is

f (x) \sim \frac{a _{0}}{2} + \sum_{n = 1}^{\infty} (a_{n} cos n x + b_{n} sin n x),

with Fourier coefficients

a_{n} = \frac{1}{π} \int_{- π}^{π} f (x) cos n x d x (n \geq 0), b_{n} = \frac{1}{π} \int_{- π}^{π} f (x) sin n x d x (n \geq 1) .

These follow by taking inner products with the individual basis vectors and using their orthogonality and squared norms.

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